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A185654 G.f.: exp( Sum_{n>=1} -sigma(3n)*x^n/n ).

%I A185654 #15 Jul 12 2018 11:09:28
%S A185654 1,-4,2,9,-9,-2,0,-5,9,9,0,-9,-1,-9,0,-1,9,9,-9,9,0,9,-5,-18,-18,9,7,
%T A185654 0,9,0,0,9,9,-18,18,-7,-9,-9,-9,9,-4,-9,-9,18,9,0,18,9,0,-9,-9,-8,-9,
%U A185654 18,-9,9,-18,1,-9,-18,9,0,18,18,0,0,9,-9,18,-9,5,-9,0,-9,-9,-9,-18,11,9
%N A185654 G.f.: exp( Sum_{n>=1} -sigma(3n)*x^n/n ).
%H A185654 Seiichi Manyama, <a href="/A185654/b185654.txt">Table of n, a(n) for n = 0..1000</a>
%F A185654 G.f.: E(x)^4/E(x^3) where E(x) = Product_{n>=1} (1-x^n). [From a formula by _Joerg Arndt_ in A182819]
%F A185654 a(n) = -(1/n)*Sum_{k=1..n} sigma(3*k)*a(n-k). - _Seiichi Manyama_, Mar 04 2017
%F A185654 Expansion of E(x) * E(x*w) * E(x/w) in powers of x^3 where w = exp(2 Pi i / 3). - _Michael Somos_, Jul 12 2018
%e A185654 G.f. = 1 - 4*x + 2*x^2 + 9*x^3 - 9*x^4 - 2*x^5 - 5*x^7 + 9*x^8 + ... - _Michael Somos_, Jul 12 2018
%t A185654 a[ n_] := SeriesCoefficient[ QPochhammer[ x]^4 / QPochhammer[ x^3], {x, 0, n}]; (* _Michael Somos_, Jul 12 2018 *)
%o A185654 (PARI) {a(n)=polcoeff(exp(sum(m=1,n,-sigma(3*m)*x^m/m)+x*O(x^n)),n)}
%o A185654 (PARI) {a(n)=local(X=x+x*O(x^n));polcoeff(eta(X)^4/eta(X^3),n)}
%Y A185654 Cf. A182819, A144613, A000203.
%Y A185654 Cf. Product_{n>=1} (1 - q^n)^(k+1)/(1 - q^(k*n)): A010815 (k=1), A115110 (k=2), this sequence (k=3), A282937 (k=5), A282942 (k=7).
%K A185654 sign
%O A185654 0,2
%A A185654 _Paul D. Hanna_, Feb 16 2011